A Construction That Produces Wallis-Type Formulas

Transcription

1 Advaces i Pure Mathematics htt://dxdoiorg/0436/am Published Olie Setember 03 (htt://scirorg/joural/am) A Costructio That Produces Wallis-Tye Formulas Joshua M Fitzhugh David L Farsorth School of Mathematical Scieces Rochester Istitute of Techology Rochester USA JMF76@ritedu DLFSMA@ritedu Received Jue 8 03; revised July 3 03 acceted August 03 Coyright 03 Joshua M Fitzhugh David L Farsorth This is a oe access article distributed uder the Creative Commos Attributio Licese hich ermits urestricted use distributio ad reroductio i ay medium rovided the origial ork is roerly cited ABSTRACT Geeralizatios of the geometric costructio that reeatedly attaches rectagles to a square origially give by Myerso are reseted The iitial square is relaced ith a rectagle ad also the dimesioality of the costructio is icreased By selectig values for the various arameters such as the legths of the sides of the origial rectagle or rectagular box i dimesios more tha to ad their relatioshis to the size of the attached rectagles or rectagular boxes some iterestig formulas are foud Examles are Wallis-tye ifiite-roduct formulas for the areas of -circles ith > Keyords: Wallis s Formula; Uit -Circle; Ifiite Product; ; Gamma Fuctio Itroductio Wallis s roduct formula for / is () To more-or-less elemetary roofs are give i [] A iterestig geometric costructio hich first aeared i [3] roduces this ifiite roduct The costructio is somehat geeralized i [45] The urose of this aer is to further geeralize the costructio i [3] Amog the aturally occurrig secial cases of the geeralizatio are ifiite-roduct reresetatios of areas of -circles I Sectio e give a accout of the costructio ad some geeralizatios I Sectio 3 e revie the gamma fuctio sice our results are ritte i terms of that fuctio Sectio 4 describes uit suer-circles or -circles x y for sice their areas are roduced by certai geeralizatios of the costructio Sectio 5 cotais several iterestig outcomes of some e geeralizatios of the geometric costructio icludig the Wallis formula for -circles The Costructio The folloig costructio roduces the Wallis roduct () [3-5] See Figure Let j be the idth ad h i be the height of the curret rectagle at the curret ste for the aroriate values of i ad j The iitial square ad the first fe stes i the costructio are: The iitial square i Figure (a) has sides 0 = ad h 0 = ad area The first ste is to attach to the right a square ith sides 0 = ad h 0 = so that the curret rectagle i Figure (b) has sides = ad h 0 = ad area The secod ste is to attach to the to a rectagle of area ith sides = ad h h 0 = / so that the curret rectagle i Figure (c) has sides = ad h = 3/ ad area 3 The third ste is to attach to the right a rectagle of area ith sides = /3 ad h = 3/ so that the curret rectagle i Figure (d) has sides = 8/3 ad h = 3/ ad area 4 The fourth ste is to attach to the to a rectagle of area ith sides = 8/3 ad h h = 3/8 so that the curret rectagle i Figure (e) has sides = 8/3 ad h = 5/8 ad area 5 The costructio cotiues idefiitely i this ay I [3-5] it is sho that h () hich is / by Wallis s roduct formula () Short [4] ad Short ad Melville [5] geeralize the costructio ith all the rectagles adjoied to the right havig area A all the rectagles adjoied to the to havig

2 580 h 0 h 0 0 (a) (b) h h (c) (d) h (e) Figure The iitial uit square ad the first four stes of the costructio area B ad the iitial square havig area 0 h 0 They sho that: If A > B the h ; if A < B the h 0 ; ad if A = B the h coverges For the last case they demostrate ho averagig methods ca be emloyed to obtai h umerically to a desired recisio These results are examles i the further geeralizatios i Sectio 5 so e do ot discuss their details The geeralizatios i Sectio 5 iclude alloig the iitial figure to be a rectagle ad icreasig the dimesioality of the costructio Proerties of the gamma fuctio are very useful ad are revieed i the ext sectio 3 The Gamma Fuctio The urose of this sectio is to remid readers about some roerties of the gamma fuctio icludig ifiite-roduct reresetatios A source that is relatively comlete ad takes a historical ersective is [6] We restrict the domai of the gamma fuctio to the ositive real umbers sice oly those values cocer us The most familiar defiitio of the gamma fuctio is the coverget imroer itegral x x e d x 0 (3) 0 hich is ko as Euler s Itegral of the Secod Kid [7 4] Itegratig (3) by arts gives (4) For ositive iteger! To secial values are ad (5) (6)

3 58 [ ] A formulatio as a limit is x x d x (7) 0 hich ca be derived from (3) [ ] Ofte Taery s theorem is cited i roofs A ifiite roduct reresetatio is! hich ca be derived by the chage of variable x = y i (7) ad erformig itegratios by arts [ ] The idetity hich is valid for a a am b b bm b b bm a a a m m ai b i i i m (8) (9) [ ] greatly simlifies may derivatios i Sectio 5 The ifiite roduct o the left-had side of (8) ould have diverged if there ere ot the same umber of terms i its umerator ad deomiator or if (9) ere ot satisfied Takig a = a = 0 ad b = b = / e obtai the Wallis roduct formula () sice 0 0 Euler s Itegral of the First Kid also ko as the beta fuctio is 0 x x d x (0) 0 0 [ ] 4 Suer Circles I the same ay that Euclidea geometry is based o the uit circle x + y = l geometries are based o the uit suer-circle or -circle x y These are Mikoski geometries hich are characterized by their uit circles that eclose a covex symmetric regio [0] See Figure for grahs of the l uit circles for = 3/ ad 4 08 y x Figure The l uit circles x + y = for = 3/ ad 4

4 58 The eclosed area of the l suer circle is 4 A 4 x d x t t dt () usig (4) ad (0) here t = x The eclosed regio is ot covex for 0 < < so the circle does ot give a Mikoski geometry; hoever () gives the areas of those regios For = () gives usig (4) (5) ad (6) The eclosed area of the uer half of the l suer circle is U A () 5 Geeralizatios of the Costructio The geeralizatios i this sectio iclude alloig the iitial figure to be a rectagle ad icreasig the dimesioality of the costructio Selectig various values for the arameters gives iterestig formulas icludig Wallis-tye formulas for half the areas of -circles 5 Startig ith a Rectagle i To Dimesios The first geeralizatio of the costructio is to begi ith a rectagle ith idth a ad height b istead of a square so that a ad h b (3) 0 0 The adjoied rectagles have areas A ad B as described i Sectio The iterative stes of adjoiig rectagles are determied by h A (4) ad h h B (5) for 0 givig the ext values of + ad h + i tur After stes the area of the hole figure hich is a rectagle is h ab A B (6) ad after + stes the area of the rectagle is h ab AB A (7) We fid recursio relatioshis betee + ad ad betee h + ad h Substitutig (6) ito (4) gives ABab A ABab A h A h A h A ab AB for 0 Substitutig (7) ito (5) gives h ABab AB ABab A h h B h Bh h B h h B ab AB A h for 0 Dividig (8) by (9) gives ab A ABab A ABab AB AB h h A B ab From (3) ad (0) ab A ab A A B A B h ab ab A B A B A B ab A ab A a A B A B h b 0 ab ab A B A B A B (8) (9) (0) I order to aly (8) the idex i the ifiite roduct must start at ot 0 Chagig the idex to m = + ad revertig back to give The h ab B ab B a A B A B () b ab A B ab A B A B ab A B ab a A B A B h ab B ab B b A B A B if ad oly if ab B ab B ab A B ab A B AB AB AB ()

5 583 from (8) ad (9) Equatio () imlies that A = values of a ad b ad usig (4) h ab a A A ab A B for ay (3) For the secial case a = b = A = B = () ad (3) is the Wallis roduct () ith () Also this is U() from () 5 Three Dimesios There is a large variety of ratios ad limits to ivestigate he the dimesioality is icreased I three dimesios the rocess is determied by the iitial box ith sides a h b ad d c (4) ad adjoiig rectagular boxes istead of rectagles The iterative stes of adjoiig rectagular boxes of volumes A B ad C are determied by ad h d A (5) h h d B (6) h d d C (7) After 3 stes the volume of the hole rectagular box is after 3 + stes the volume is hd abc AB C (8) hd abc ABC A (9) ad after 3 + stes the volume is hd abc ABC A B (30) Solvig (5) for ad usig ( 8) gives ABCabc A h d A h d A abc ABC A B C abc A (3) Similarly (6) ad (9) give h ABC abc A B h (3) AB C abc A ad (7) ad (30) give d ABC abc ABC d (33) ABC abc AB s of ratios of (3) (3) ad (33) give a variety of iterestig exressios We reset three examles for /d Dividig (3) by (33) gives AB C abc A d d AB C abc AB C abc A B (34) AB C abc A B C abc A abc A B A B C A B C d abc abc A B C A B C A B C From (4) ad (34) abc A abc A B a A B C A B C d c 0 abc abc A B C A B C A B C Chagig the idex to m = + ad revertig back to give d abc B C abc C (35) a A B C A B C c abc A B C abc A B C A B C The abc A B C a A B C d abc B C c A B C abc A B C abc C A B C if ad oly if abc B C abc C A B C A B C (36) abc A B C abc A B C A B C from (8) ad (9) Equatio (36) imlies that A = C for ay values of a b c ad B ad usig (4) a abc d c A B abc (37) A B abc A abc A B AB AB

6 584 Cosider three secial cases of (37) For the first oe set a = b =c = A = B = C = the () (35) ad (37) give 3 3 d U 3 For the secod secial case set (38) A B abc a = c ad A = abc i (35) ad (37) to obtai d U (39) for > The domai of U() is restricted by A B abca B A B A For the third case set A B abc a = c ad A + B = abc i (35) ad (37) to obtai d U (40) for < < The domai is restricted by AB abcab AB B A B Usig this costructio Wallis-tye roduct formulas have bee obtaied for half the areas of -circles for > 53 Dimesios For the costructio i dimesios e limit the iitial shae to be a uit hyercube ad each adjoied shae to be rectagular of uit hyer volume The th value of the i th side s legth is s (i) ad the iitial sides legths are s 0 ( i) = ith i Aalogous to (4) ad (5) i Subsectio 5 for to dimesios ad (5) ( 6) ad (7) i Subsectio 5 for three dimesios the iterative stes of adjoiig rectagular boxes are determied by i s s s i for the ( + ) th ste j s i sj sj s i i i j for the ( + j) th ste ith < j < ad i s i s s for the ( + ) th ste Aalogous to (6) ad (7) i Subsectio 5 for to dimesios ad (8) (9) ad (30) i Subsectio 5 for three dimesios the volumes are s i i after the () th ste ad j s i s i j i i j after the ( + j) th ste ith j Usig a aalysis arallelig Subsectios 5 ad 5 e obtai s u u v s v v u u v v u for all u ad v equal to For u = ad v = s s U (4) For = (4) is the Wallis formula () ith () ad for = 3 it is (38) 6 Summary We have geeralized the ifiite geometric costructio of attachig rectagles to a square hich as origially reseted i [3] by alloig the iitial square to be relaced ith a rectagle ad by icreasig the dimesioality of the costructio Selectig values for the various arameters such as the legths of the sides of the origial rectagle or rectagular box i dimesios more tha to ad their relatioshis to the size of the attached rectagles or rectagular boxes gives some iterestig formulas Examles are Wallis-tye formulas (38) through

7 585 (4) for half the areas of -circles ith > REFERECES [] S J Miller A Probabilistic Proof of Wallis s Formula for The America Mathematical Mothly Vol 5 o [] J Wästlud A Elemetary Proof of the Wallis Product Formula for Pi The America Mathematical Mothly Vol 4 o [3] G Myerso The ig Shae of a Sequece of Rectagles The America Mathematical Mothly Vol 99 o doi:0307/35077 [4] L Short Some Geeralizatios of the Wallis Product Iteratioal Joural of Mathematical Educatio i Sciece ad Techology Vol 3 o doi:0080/ [5] L Short ad J P Melville A Uexected Aearace of Pi Mathematical Sectrum Vol 5 o [6] G K Sriivasa The Gamma Fuctio: A Eclectic Tour The America Mathematical Mothly Vol 4 o [7] E T Whittaker ad G Watso A Course i Moder Aalysis 3rd Editio Cambridge Uiversity Press Cambridge 90 [8] T M Aostol Mathematical Aalysis Addiso-Wesley Readig 957 [9] T J Bromich A Itroductio to the Theory of Ifiite Series d Editio Revised Macmilla Lodo 96 [0] A C Thomso Mikoski Geometry Cambridge Uiversity Press Cambridge 996 doi:007/cbo